Universal driving protocol for symmetry-protected Floquet topological phases

Abstract

We propose a universal driving protocol for the realization of symmetry-protected topological phases in 2 + 1 dimensional Floquet systems. Our proposal is based on the theoretical analysis of the possible symmetries of a square lattice model with pairwise nearest-neighbor coupling terms. Among the eight possible symmetry operators we identify the two relevant choices for topological phases with either time-reversal, chiral, or particle-hole symmetry. From the corresponding symmetry conditions on the protocol parameters, we obtain the universal driving protocol where each of the symmetries can be realized or broken individually. We provide specific parameter values for the different cases, and demonstrate the existence of symmetry-protected copropagating and counterpropagating topological boundary states. The driving protocol especially allows us to switch between bosonic and fermionic time-reversal symmetry, and thus between a trivial and non-trivial symmetry-protected topological phase, through continuous variation of a parameter. TRS CS PHS Π 2 = 1 step 1 ∆A = ∆B = ∆ ∆A = ∆ ∆C = ∆D = −∆ ∆C = −∆ this J = 2π/T J = 2π/T J = 2π/T work ∆ = 3/T ∆ = 9/T J = π/T

Real and imaginary edge states in stacked Floquet honeycomb lattices

“…In our model, we add a second honeycomb layer on which an inverse copy of the driving protocol is implemented, similiar to the procedure in references [10,33]. The two layers (indicated by red circles and blue diamonds in Fig.…”

“…It was recognized only recently that non-Hermiticity extends this picture even further, both for static [18][19][20][21][22] and Floquet systems [23][24][25][26]. In addition to the familiar topological phases with real band gaps, new non-Hermitian topological phases emerge [27][28][29] which arise from imaginary and point gaps in the complex-valued spectrum.…”

Controlling the direction of topological transport in a non-Hermitian time-reversal symmetric Floquet ladder

“…In lattices of evanescently coupled waveguides 11 , spatially periodic modulation of the waveguides along the propagation direction has been used to achieve the encoding of gauge fields and fermionic degrees of freedom via Floquet engineering [12][13][14][15][16] . For example, helical waveguide arrays mimic the interaction of electrons in a solid with an external magnetic field 17 , and Floquet protocols with pairwise coupling can faithfully reproduce the Kramers degeneracy of timereversal symmetric spin 1/2 electrons 10,18 .…”

Non-Hermitian Pseudo-Gaps

“…5(e). With a nonzero t 2 , the edge modes partially merge into the bulk to become counter-propagating chiral modes [88][89][90][91][92][93][94]. Interestingly, well-defined edge modes can also traverse a pseudo-gap even if the Chern number is not welldefined, such as the µ = 1 case at the topological transition.…”